Global safety formats in fib Model Code 2010 for design of concrete structures

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1 Cervenka, V: Proceeings of the 11 th International Probabilistic Workshop, Brno 2013 Global safety forats in fib Moel Coe 2010 for esign of concrete structures Vlaiir Cervenka Cervenka Consulting, Prague, Czech epublic Abstract: Moel Coe 2010 introuces non-linear analysis in esign of concrete structures requires an alternative approach to safety verification an relate global safety forat. Several ethos for verification of liit states using non-linear analysis are presente: full probabilistic etho, etho ECOV base on the estiate of resistance variation, global safety factor accoring to EN an partial safety factors. The ethos are copare on several exaples of reinforce concrete structures ranging fro uctile to brittle oes of failure. Keywors: non-linear analysis, safety forats, fracture echanics. 1 Introuction The new fib Moel Coe 2010 [1] evelope within the international scientific counity represents the state-of-the art for esign of concrete structures. It reveals trens an ieas for future coe evelopent while it is an operational coe, useful for practical esign. One of the new features is the introuction of a global safety forat propose for verification of resistance assiste by non-linear analysis. This opens possibilities for nuerical siulation base on nonlinear analysis to be use as tool in esign process. Such innovations observe in other inustrial branches are now spreaing also to concrete inustry. In a stanar esign process the loa actions are eterine for chosen critical cross sections by elastic structural analysis. They represent a possible istribution of internal forces satisfying equilibriu conition, but o not reflect a force reistribution ue to nonlinear effects. The safety verification is ae in local points whereas a global safety of the whole structure is not evaluate. This eficiency can be reove by applying nonlinear analysis. A non-linear structural analysis base on realistic constitutive relations akes possible a siulation of a real structural behavior. It reflects an integral response, where all local points (sections) interact an therefore it requires an aequate approach for safety

2 Cervenka,V.: Global safety forats in fib Moel Coe 2010 for esign of concrete structures verification. The non-linear analysis offers a verification of global resistance an requires a safety forat for global resistance. Several ethos for verification of liit states using non-linear analysis are presente: full probabilistic etho, etho ECOV propose by the author, global safety factor accoring to EN an partial safety factors. These ethos are base on a coon probabilistic safety concept for verification of liit states. They iffer in the level of ipleentation of the probability ethos. The ethos are copare on several exaples of reinforce concrete structures ranging fro uctile to brittle oes of failure. 2 Global safety forats 2.1 Global esign conition For the verification of resistance the esign conition can be approxiate by the inequality where the extree values of actions an resistance are ecouple as follows: F < (1) It this F is esign action an is esign resistance an both these entities cover safety argins. The safety of loaing an resistance are treate separately, which is a certain approxiation as copare to a general probabilistic approach. In esign practise base on the partial safety factors this siplification is accepte. F = F( S, γ, γ, γ,..), where the G Q P representative loa S is factorize by partial safety factors γ G, γ Q, γ P,.. for peranent loa, live loa, pre-stressing, etc. In nonlinear analysis, represents the global resistance in ters of forces corresponing to actions (live loa, horizontal loa, etc.). Note, that in partial safety factor etho we assue a failure probabilities of separate aterials, but o not evaluate the failure probability on the structural level. Unlike in sectional esign, the global resistance reflects an integral response of the whole structure, in which all aterial points (or cross sections) interact. The safety argin can be expresse by the safety factor: = (2) γ where is the ean resistance. (This is soeties referre to as noinal resistance.) The global safety factor γ covers all uncertainties an can be relate to the coefficient of variation of resistance V (assuing a log-noral istribution, accoring to Eurocoe 2) as γ = exp( α β V ) where α is the sensitivity factor for resistance an β is the reliability inex. It is recognize that variability inclue in V epens on uncertainties ue to various sources: aterial properties, geoetry an resistance oel. They can be treate as rano effects an analyze by probabilistic ethos. Due to available statistical ata the probabilistic treatent of aterials an geoetry can be one in a rational way. However, a rano (3)

3 Cervenka, V: Proceeings of the 11 th International Probabilistic Workshop, Brno 2013 treatent of oel uncertainties is ore ifficult, because of liite ata. A siplifie forulation was propose in MC2010, where in enoinator of the right han sie in Eq.(2) is a prouct of two factors γ = γ γ. (It follows fro eterination of partial safety factors in MC2010, Sect ). The first factor γ is relate to aterial uncertainty an can be establishe by a probabilistic analysis. The secon factor γ is relate to oel an geoetrical uncertainties an recoene values are in the range (as suggeste by Eurocoe 2-2.) ecent investigation by Schlune et.al. [7] foun such values unsafe an propose a ore general etho in which the overall coefficient of resistance variation can be eterine as V = V + V + V (4) G Where variability ue to specific sources are ientifie: V G - geoetry, V - aterial strength, V - oel. This approach allows to inclue all uncertainties in ore rational way. Base on a survey of various blin bench ark stuies Schlune conclue that oel uncertainties of nonlinear analysis are uch higher than in stanar esign base on engineering forulas an are strongly epenent on oes of failure. eporte coefficients of variation ue to oel uncertainty for bening failure in range 5-30%, for shear 15-64%. Schlune conclue that ue to the lack of ata, the choice of the oel uncertainty often epens on engineering jugent an can be subjective. However, these conclusions o not recognize the effect of oel valiation, which can ecrease oel uncertainties. Further research is neee to recoen appropriate values of the oel uncertainty for nuerical siulations. The assessent of the safety accoring to Eq.(1) can be one by various ethos, ranging fro a full probabilistic analysis to the partial factor etho, which iffer in the level of approxiations involve. They will be briefly escribe in the next sections. 2.2 Full probabilistic analysis The probabilistic analysis is the ost rational tool for the safety assessent of structures. It can be further refine by introucing non-linear structural analysis as a liit state function. The nuerical siulation resebles a real testing of structures by consiering a representattive group of saples, which can be statistically analyze for assessent of safety. We shall only briefly outline an approach ipleente in the software tool SAA [3]. More about the probabilistic analysis can be foun in [6]. In nuerical siulations the probabilistic analysis of resistance can be perfore by LHS etho, in which the aterial input paraeters are varie in a systeatic way. The resulting array of resistance values is approxiate by a istribution function of global resistance an escribes the rano variation of resistance. Finally, for a require reliability inex β, or probability of failure P f, the value of esign resistance shall be calculate. The probabilistic analysis base on nuerical siulation with rano sapling can be briefly escribe as follows:

4 Cervenka,V.: Global safety forats in fib Moel Coe 2010 for esign of concrete structures Forulation of a nuerical oel base on non-linear finite eleent etho. Such a oel escribes the resistance function an can perfor eterinistic analysis of resistance for a given set of input variables. anoization of input variables (aterial properties, iensions, bounary conitions, etc.). This can also inclue soe effects, which are not in the action function (for exaple pre-stressing, ea loa etc.). ano aterial properties are efine by a rano istribution type an its paraeters (ean, stanar eviation, etc.). They escribe the uncertainties ue to variation of resistance properties. The ranoization can be one by two ethos: (1) ano variables, where the paraeter is constant within a saple (structure), but changes between saples. (2) ano fiels, where the paraeter is ranoly variable within a saple. A correlation of rano variables shoul be consiere appropriately. Probabilistic analysis of resistance. This can be perfore by the nuerical etho of Monte Carlo-type of sapling, such as the LHS sapling etho. esults of this analysis provie rano paraeters of resistance, such as ean, stanar eviation, etc. an the type of istribution function for resistance (PDF). Evaluation of esign resistance base on the reliability inex β or probability of failure. In this a esign point is foun by extrapolation of point aroun central region base on PDF. The avantage of a full probabilistic analysis is that it is inepenent of a failure oe. A potentially higher safety argins of soe failure oes, such as for exaple shear failure, is autoatically inclue in higher sensitivity of nuerical resistance to a brittle failure. A isavantage of this approach is in the fact that the target value of esign resistance is locate in the tail of probability istribution function (PDF), which is eterine by the best fit fro the sapling. The esign value is obtaine by extrapolation an strongly epens on the choice of PDF. On the other han the approach is nuerically robust, coputationally efficient an feasible for practical application. However, ue to its coputational eans a full probabilistic analysis is justifie in special cases, where consequences of failure justify the effort. 2.3 ECOV etho estiate of coefficient of variation A siplifie probabilistic analysis was propose by the author [4] in which the rano variation of resistance is estiate using only two saples. It is base on the iea, that the rano istribution of resistance, which is escribe by the coefficient of variation V, can be estiate fro ean an characteristic values k of resistance. The unerlying assuption is that rano istribution of resistance is accoring to a lognoral istribution, which is typical for structural resistance. In this case, it is possible to express the coefficient of variation as: V 1 ln = 1.65 k (5) The global safety factor γ of resistance is then estiate by Eq.(3) using the typical values β = 3.8 (50 years) an α = 0.8 (which correspons to the failure probability P f

5 Cervenka, V: Proceeings of the 11 th International Probabilistic Workshop, Brno 2013 =0.001). The global resistance factor can be irectly relate to the estiate coefficient of variation V as γ exp(3.04 V ) an the esign resistance is obtaine fro Eq.(2). The keystone in this etho is the eterination of the ean an characteristic values of the resistance:,. It is propose to estiate the by two separate nonlinear analyses k using ean an characteristic values of input aterial paraeters, respectively. The etho is general an the safety escribe by the reliability inex β can be change if require. Also the istribution function PDF can be change if justifie. It reflects all types of failure. The sensitivity to rano variation of all aterial paraeters is autoatically inclue. Thus, there is no nee of special oifications of concrete properties in orer to copensate for greater rano variation of certain properties as in the next etho EN A siilar an refine etho with ore saples was propose by Schlune et al.[6] 2.4 Metho base on EN Eurocoe 2 for briges introuce a concept for global safety verification base on nonlinear analysis. Design resistance is calculate fro f =, f = ( f y, f...) (6) γ Where f, f are ean values of aterial paraeters of steel reinforceent an concrete, f y y = 1.1 f an f = 0.85 f. The global factor of resistance shall be γ = 1, 27. yk esistance value ck f is not a ean value if the concrete fails. The concrete strength f is reuce to account for higher variability of concrete property. Fig. 1 Probabilistic efinition of ean-, characteristic-k an esign- values for steel an concrete failure, f-reuce concrete strength.

6 Cervenka,V.: Global safety forats in fib Moel Coe 2010 for esign of concrete structures A justification for the introuction of concrete strength paraeter f is base on the iea of reflecting the safety of partial safety factors in the newly introuce global safety factor. The concept illustrate in Error! eference source not foun., where probability ensity functions for both aterials are copare. The strength paraeters on horizontal axis are noinal with respect to esign values. It is assue that esign values for concrete an steel correspon to the sae probability. They are locate at point 1. on the horizontal axis. In steel the esign value f y = f yk / γ s is erive fro the characteristic strength f yk with the use of partial safety factor γ s = 1.15, the ean value is assue to be f y = 1.1 f yk, which leas to a safety factor 1.27 with respect to ean. In concrete the esign value fc = fck / γ c, is erive fro the characteristic strength f ck with the use of partial safety factor γ c = 1.5. We introuce a new paraeter for concrete f, which correspon to the safety factor of steel for en, an is locate at the value 1.27 on the horizontal axis in Error! eference source not foun.. fck f = γ fc = γ s fck (6) γ c It shoul be note that the value of strength f oes not represent a ean value. Instea, it is a value corresponing to a lower probability than characteristic value an inclues the aitional safety require for concrete as copare to steel. The subject is also treate by Bertagnoli et al. [8]. The avantage of the above etho is, that it covers both oels of failure, ue to steel an concrete, without necessity of a prior knowlege of failure oe. For concrete the Eurocoe 2 allows only copressive type of failure an exclues failure types relying of tension. This, of course, prevents a wie range of applications, such as shear, or pull-out of fastenings. The stuy presente in [2] extens its applications also to brittle oes of failure. 2.5 Partial safety factors (PSF) The etho of partial safety factors, which is use in ost esign coes can be irectly applie to global analysis in orer to obtain the esign resistance = ( f ). In this, the esign values of aterial paraeters f = fk / γ M are use for analysis input ( fk are characteristic values an γ M partial safety factors of aterials). It can be argue, that esign values represent extreely low aterial properties, which o not represent a real aterial behavior an thus can lea to istorte failure oes. On the other han, this etho aresses irectly the target esign value an thus no extrapolation is involve. However, the probability of global resistance is not evaluate an therefore not known.

7 2.6 Coparison of concepts Cervenka, V: Proceeings of the 11 th International Probabilistic Workshop, Brno 2013 The ethos outline above offer an estiation of esign resistance with various levels of approxiations. The full probabilistic analysis is regare here as the ost rational as suggeste by the Joint Coittee for Structural safety. The other ethos are approxiations base on siplifying assuptions, which allow estiation of resistance esign values. Brief suary of the ethos is show in Table 1. an the probabilistic concept of these forats is illustrate in Error! eference source not foun.. Fig. 2 Probabilistic concept of global safety forats. Table 1 Suary of ethos for verification of global safety Metho Material paraeters require equire nuber of resistance calculations Approxiation target (1) Probabilistic (LHS sapling) Probability istribution Depens on nuber of saples (8-30) exact (2) ECOV characteristic, ean 2 variability of resistance (3) EN characteristic 1 (4) PSF esign 1 ean esign It has been observe that the safety of resistance epens not only the variation of basic aterial paraeters, but also to the oe of failure. In other wors, for the sae concrete aterial, structures with ifferent type of failure can have ifferent variability of resistance. In this respect the ost rational approach is by the full probabilistic forat (1), in which the rano istribution of resistance is eterine an the esign value of resistance is chosen for a require probability of failure (an reliability inex β ). The other three ethos (2,3,4 in Table 1) can be regare as approxiate fro the probabilistic point of view. The concept of the Metho (2) is very close to a probabilistic forat, since it works with the variance of resistance an calculations with ean an characteristic paraeters are relatively robust. Metho (3) is using a unique global safety factor. Assessent of resistance near ean is relatively robust an an effect of concrete variability is inclue in the reuce concrete strength. Metho (4) by partial safety factors, offers a irect estiate of esign value without a nee of estiating global safety

8 Cervenka,V.: Global safety forats in fib Moel Coe 2010 for esign of concrete structures argin. In conclusion each etho has its erit an, as will be shown later none sees to be superior to the others. The author has initiate investigations with the ai to copare the various safety forats [2],[4]. The stuy coprise of a wie range of structures incluing: siple bea, laboratory test of shear wall, laboratory test of a eep bea, in-situ test of a real structure brige an a esign case of a brige pier, SFC concrete. A variety of failure oes covere uctile bening oe, brittle shear oes an a concrete copression oe. Details of this investigation can be foun in [2]. A suary of results is shown in Table 2. Three approxiate ethos, naely the partial safety factors (PSF), etho base on estiate of coefficient or variation of resistance (ECOV) an etho accoring to EN are evaluate. The table shows the ratio of resistances foun by approxiate ethos to the full probabilistic analysis (which is consiere as ost exact for this purpose). It is note that the stuy oes not reflect the oel uncertainty in a consistent way. The ethos PSF an EN inclue the oel uncertainty as given by Eurocoe, while the ECOV an full probabilistic analysis it is not consiere in orer to siplify the coparison. This can explain the average results of ECOV etho being slightly higher than the other two ethos. Table 2 Suary of ethos for verification of global safety. / prob. PSF ECOV EN Exaple 1 Bening Exaple 2 eep bea Exaple 3 brige pier Exaple 4 brige frae Exaple 5 shear bea Y Exaple 6 shear bea Y average The stuy confire feasibility of the approxiate ethos for the safety assessent. The etho ECOV is preferre since it relates the safety to the resistance rano variation an is consiere ore rational as copare to EN etho. Multiple failure oes, which are typical features of reinforce concrete structures are inherently inclue in the nuerical oels an thus they are reflecte in results of analysis an resistance variability. Therefore, the approxiate ethos of safety verification are generally applicable in esign. In significant cases, if justifie by failure consequences, a full probabilistic analysis shoul be applie.

9 Cervenka, V: Proceeings of the 11 th International Probabilistic Workshop, Brno Application For illustration an application of esign verification by nonlinear analysis will be shown. The exaple is a large bea teste in laboratory by Yoshia an Collins [8]. The size of the bea is large an excees usual bea iensions (span=12, epth=2) an has no vertical reinforceent. The shear failure is apparently influence by its large size an is very brittle. The failure oe was well capture by the nuerical siulation as illustrate in Fig. 3. Coparison of resistances obtaine by various safety forats is shown in Fig. 4Furtherore it shows the values of esign resistance by coes EN an ACI 318. Fig. 3 Nuerical an experiental crack pattern. Fig. 4 Design resistance of large bea by Yoshia [8] accoring to various safety forats an coes. This case ha shown two rearkable features of nuerical siulation. First, a refine constitutive oeling base on fracture echanics can capture the size effect of brittle shear failure an provies a ore safe oel of resistance. Secon, the global safety forats offer consistent safety argins for the esign verification. 4 Closing rearks Moel Coe 2010 introuce a verification assiste by nuerical siulation as one of the esign ethos an a global safety forat. The of application is extene beyon the scope of engineering ethos base on elastic istribution of internal forces in cross

10 Cervenka,V.: Global safety forats in fib Moel Coe 2010 for esign of concrete structures sections into nonlinear analysis. Due to its general approach it overcoes the liits of stanar esign base on beas an coluns. On the other han it introuces potentially higher oel uncertainties. Therefore the oel valiation becoes an iportant requireent for its application in engineering practice. The fib Moel Coe 2010 outlines the fraework of liit state verification by nuerical siulations an introuces the global safety forats suggeste for this purpose. Further research is neee in orer to iprove the guie for valiation of nuerical oels an for the classification of oel uncertainties. Acknowlegeent The paper is base in part on the research project by the Czech Science Founation no. P105/12/2051. The financial support is greatly appreciate. Literature [1] Walraven, J., (eitor). fib Moel Coe for Concrete Structures 2010 Septeber ISBN: (See also fib Bulletin 65 an 66.) [2] Cervenka, V., eliability-base non-linear analysis accoring to fib Moel Coe 2010, Structural Concrete, Journal of the fib, Vol. 14, March 2013, ISSN , pp , DOI: /suco [3] Cervenka V. (eitor): SAA Structural Analysis an eliability Assessent. User s anual. Cervenka Consulting, Prague [4] Cervenka V. : Global Safety Forat for Nonlinear Calculation of einforce Concrete. Beton- un Stahlbetonbau 103 (2008), Special Eition, Ernst&Sohn. pp [5] Cervenka, V., Dolezel, J., Novak, D, "Shear Failure of Large Lightly einforce Concrete Beas: PAT II Assessent of Global Safety of esistance". Proc. 3r fib International Congress 2010, May 29-June 2, 2010, Washington DC, USA. [6] Vrouvenweler, A.C.W.M.: Conclusions of the JCSS Workshop on Sei-probabilistic FEM calculations, Delft, 1-2 Deceber [7] Schlune, H., Plos, M., Gylltoft, K.: Safety Forats for Nonlinear Analysis of Concrete Structures. Engineering Structures, Elsewier, Vol.33,No.8,August [8] Bertagnoli, G., Giorano L., Mancini, G. (2004): Safety foratfor the nonlinear analysis of concrete structures. STUDIES AND ESEACHES V.25, Politechnico i Milano, Italy. [9] Yoshia Y.: Shear einforceent for Large Lightly einforce Concrete Mebers, MS Thesis uner supervision of Prof. Collins, 2000, Univ. of Toronto, Canaa.